Bias and Variability Lecture 38 Section 8.3 Wed, Mar 31, 2004

Unbiased Statistics A statistic is unbiased if its average value equals the parameter that it is estimating. A statistic is unbiased if its average value equals the parameter that it is estimating. The variability of a statistic is a measure of how spread out its sampling distribution is. The variability of a statistic is a measure of how spread out its sampling distribution is. All estimators exhibit some variability. All estimators exhibit some variability.

The Parameter the parameter

Unbiased, Low Variability the parameter

Unbiased, High Variability the parameter

Biased, High Variability the parameter

Biased, Low Variability the parameter

Accuracy and Precision An unbiased statistic allows us to make accurate estimates. An unbiased statistic allows us to make accurate estimates. A low variability statistic allows us to make precise estimates. A low variability statistic allows us to make precise estimates. The best estimator is one that is unbiased and with low variability. The best estimator is one that is unbiased and with low variability.

Experiment Me: randBin(50,.1, 200) Me: randBin(50,.1, 200) Volunteer #1: randBin(50,.3, 200) Volunteer #1: randBin(50,.3, 200) Volunteer #2: randBin(50,.5, 200) Volunteer #2: randBin(50,.5, 200) Volunteer #3: randBin(50,.7, 200) Volunteer #3: randBin(50,.7, 200) Volunteer #4: randBin(50,.9, 200) Volunteer #4: randBin(50,.9, 200) It will take the TI-83 about 6 minutes. It will take the TI-83 about 6 minutes.

Experiment Divide the list by 50 to get proportions. Divide the list by 50 to get proportions. Store the results in list L 1. Store the results in list L 1. STO L 1. STO L 1. Compute the statistics for L 1. Compute the statistics for L 1. 1-Var Stats L 1. 1-Var Stats L 1. What are the mean and standard deviation? What are the mean and standard deviation?

The Sampling Distribution of p^ The distribution of p^ is approximately normal. The distribution of p^ is approximately normal. The approximation is excellent if The approximation is excellent if np ≥ 5 and n(1 – p) ≥ 5. The mean is p (the pop. proportion). The mean is p (the pop. proportion). The standard deviation is The standard deviation is  (p(1 – p)/n).

The Sampling Distribution of p^ Because n appears in the denominator of the standard deviation, Because n appears in the denominator of the standard deviation, The standard deviation decreases as n increases. The standard deviation decreases as n increases. However, it does not decrease as fast as n increases. However, it does not decrease as fast as n increases. For example, if the sample is 100 times larger, the standard deviation is only 10 times smaller. For example, if the sample is 100 times larger, the standard deviation is only 10 times smaller.

Example A recent poll showed that 44% of the voters favor George W. Bush in the 2004 election. A recent poll showed that 44% of the voters favor George W. Bush in the 2004 election. Suppose that the population proportion p = Suppose that the population proportion p = Describe the distribution of p^ if the sample size is n = Describe the distribution of p^ if the sample size is n = 1000.

Example If we take a sample of 1000 voters, what is the probability that at least 50% of them favor Bush? If we take a sample of 1000 voters, what is the probability that at least 50% of them favor Bush? The z-score of 0.50 is z = The z-score of 0.50 is z = P(p^ > 0.50) = P(Z > 3.82) P(p^ > 0.50) = P(Z > 3.82) =

Example Find the probability that less than 42% favor Bush. Find the probability that less than 42% favor Bush. Find the probability that between 42% and 46% favor Bush. Find the probability that between 42% and 46% favor Bush.

Let’s Do It! Let’s do it! 8.5, p. 484 – Probabilities about the Proportion of People with Type B Blood. Let’s do it! 8.5, p. 484 – Probabilities about the Proportion of People with Type B Blood. Let’s do it! 8.6, p. 485 – Estimating the Proportion of Patients with Side Effects. Let’s do it! 8.6, p. 485 – Estimating the Proportion of Patients with Side Effects. Let’s do it! 8.7, p. 487 – Testing hypotheses about Smoking Habits. Let’s do it! 8.7, p. 487 – Testing hypotheses about Smoking Habits.

Assignment Page 488: Exercises 1 – 15. Page 488: Exercises 1 – 15.